Average Error: 60.4 → 60.4
Time: 10.5s
Precision: binary64
\[-1 < \varepsilon \land \varepsilon < 1\]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(e^{\varepsilon \cdot a} - 1\right) \cdot \left(\left(\sqrt[3]{-1 + e^{\varepsilon \cdot b}} \cdot \sqrt[3]{-1 + e^{\varepsilon \cdot b}}\right) \cdot \sqrt[3]{\sqrt[3]{-1 + e^{\varepsilon \cdot b}} \cdot \left(\sqrt[3]{-1 + e^{\varepsilon \cdot b}} \cdot \sqrt[3]{-1 + e^{\varepsilon \cdot b}}\right)}\right)}\]
\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}
↓
\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(e^{\varepsilon \cdot a} - 1\right) \cdot \left(\left(\sqrt[3]{-1 + e^{\varepsilon \cdot b}} \cdot \sqrt[3]{-1 + e^{\varepsilon \cdot b}}\right) \cdot \sqrt[3]{\sqrt[3]{-1 + e^{\varepsilon \cdot b}} \cdot \left(\sqrt[3]{-1 + e^{\varepsilon \cdot b}} \cdot \sqrt[3]{-1 + e^{\varepsilon \cdot b}}\right)}\right)}
(FPCore (a b eps)
 :precision binary64
 (/
  (* eps (- (exp (* (+ a b) eps)) 1.0))
  (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0))))
↓
(FPCore (a b eps)
 :precision binary64
 (/
  (* eps (- (exp (* eps (+ a b))) 1.0))
  (*
   (- (exp (* eps a)) 1.0)
   (*
    (* (cbrt (+ -1.0 (exp (* eps b)))) (cbrt (+ -1.0 (exp (* eps b)))))
    (cbrt
     (*
      (cbrt (+ -1.0 (exp (* eps b))))
      (* (cbrt (+ -1.0 (exp (* eps b)))) (cbrt (+ -1.0 (exp (* eps b)))))))))))
double code(double a, double b, double eps) {
	return (eps * (exp((a + b) * eps) - 1.0)) / ((exp(a * eps) - 1.0) * (exp(b * eps) - 1.0));
}
↓
double code(double a, double b, double eps) {
	return (eps * (exp(eps * (a + b)) - 1.0)) / ((exp(eps * a) - 1.0) * ((cbrt(-1.0 + exp(eps * b)) * cbrt(-1.0 + exp(eps * b))) * cbrt(cbrt(-1.0 + exp(eps * b)) * (cbrt(-1.0 + exp(eps * b)) * cbrt(-1.0 + exp(eps * b))))));
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original60.4
Target15.0
Herbie60.4
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 60.4

    \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
  2. Using strategy rm

  3. Applied add-cube-cbrt_binary6460.4

    \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \color{blue}{\left(\left(\sqrt[3]{e^{b \cdot \varepsilon} - 1} \cdot \sqrt[3]{e^{b \cdot \varepsilon} - 1}\right) \cdot \sqrt[3]{e^{b \cdot \varepsilon} - 1}\right)}}\]

  4. Simplified60.4

    \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(\color{blue}{\left(\sqrt[3]{-1 + e^{b \cdot \varepsilon}} \cdot \sqrt[3]{-1 + e^{b \cdot \varepsilon}}\right)} \cdot \sqrt[3]{e^{b \cdot \varepsilon} - 1}\right)}\]

  5. Simplified60.4

    \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(\left(\sqrt[3]{-1 + e^{b \cdot \varepsilon}} \cdot \sqrt[3]{-1 + e^{b \cdot \varepsilon}}\right) \cdot \color{blue}{\sqrt[3]{-1 + e^{b \cdot \varepsilon}}}\right)}\]
  6. Using strategy rm

  7. Applied add-cube-cbrt_binary6460.4

    \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(\left(\sqrt[3]{-1 + e^{b \cdot \varepsilon}} \cdot \sqrt[3]{-1 + e^{b \cdot \varepsilon}}\right) \cdot \sqrt[3]{\color{blue}{\left(\sqrt[3]{-1 + e^{b \cdot \varepsilon}} \cdot \sqrt[3]{-1 + e^{b \cdot \varepsilon}}\right) \cdot \sqrt[3]{-1 + e^{b \cdot \varepsilon}}}}\right)}\]

  8. Final simplification60.4

    \[\leadsto \frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(e^{\varepsilon \cdot a} - 1\right) \cdot \left(\left(\sqrt[3]{-1 + e^{\varepsilon \cdot b}} \cdot \sqrt[3]{-1 + e^{\varepsilon \cdot b}}\right) \cdot \sqrt[3]{\sqrt[3]{-1 + e^{\varepsilon \cdot b}} \cdot \left(\sqrt[3]{-1 + e^{\varepsilon \cdot b}} \cdot \sqrt[3]{-1 + e^{\varepsilon \cdot b}}\right)}\right)}\]

Reproduce

herbie shell --seed 2020288 
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :precision binary64
  :pre (and (< -1.0 eps) (< eps 1.0))

  :herbie-target
  (/ (+ a b) (* a b))

  (/ (* eps (- (exp (* (+ a b) eps)) 1.0)) (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0))))